Is the exhaust fan necessary?

It is worth checking if your BIOS has options to shut down the PC based on the temperature of the CPU. This means if you do decide to run without the exhaust fan then your PC would shut itself down if things were getting too hot preventing damage to the components.

And in reply to your bracketed comment, I would say it's really worth looking into replacing the case fan. The difference in noise between stock components and third-party replacements costing only a few dollars can be huge. Another option would be a fan speed controller so you can lower the speed of the case fan so that it is much quieter but still provides some air flow.

This is a good article on airflow in a computer case.

The CPU is only one component of several in your case that need to stay cool. What about HD, GPU, Motherboard (North & South Bridge)? I would monitor the temps for those as well. Considering the article on airflow listed in the other post, I would just get a good, quiet rear fan. Heat kills.